Sheaves that fail to represent matrix rings

TL;DR

This paper demonstrates fundamental obstructions to representing noncommutative rings as sheaves, showing that certain natural sheaf-theoretic constructions cannot exist even when restricted to matrix rings.

Contribution

It proves two no-go theorems showing the impossibility of representing noncommutative rings via sheaves with properties analogous to the commutative case.

Findings

No subcanonical coverage includes all Zariski covers for rings.

No functor from rings to ringed categories can extend the spectrum functor to noncommutative rings.

Abstract

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go results are proved by restricting attention to matrix rings.